The radius of the incircle of the equilateral triangle having each side 6 cm is

Options

A. A)`2sqrt(3)` cm
B. B)`sqrt(3)` cm
C. C)`6sqrt(3)` cm
D. D)`2 `cm

Answer

To find the radius of the incircle of an equilateral triangle with each side measuring 6 cm, we can use the formula for the inradius \( r \) of an equilateral triangle, which is given by:

\[ r = \frac{a}{2\sqrt{3}} \]

where \( a \) is the length of a side of the triangle.

### Step-by-step solution:

1. **Identify the side length of the triangle**:
The side length \( a \) of the equilateral triangle is given as 6 cm.

2. **Substitute the side length into the formula**:
We will substitute \( a = 6 \) cm into the formula for the inradius:
\[ r = \frac{6}{2\sqrt{3}} \]

3. **Simplify the expression**:
First, simplify the fraction:
\[ r = \frac{6}{2\sqrt{3}} = \frac{6 \div 2}{\sqrt{3}} = \frac{3}{\sqrt{3}} \]

4. **Rationalize the denominator**:
To simplify \( \frac{3}{\sqrt{3}} \), we can multiply the numerator and the denominator by \( \sqrt{3} \):
\[ r = \frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3} \]

5. **Conclusion**:
The radius of the incircle of the equilateral triangle is \( \sqrt{3} \) cm.

### Final Answer:
The radius of the incircle is \( \sqrt{3} \) cm.